Sumer (a region of Mesopotamia, modern-day Iraq) was the birthplace of writing, the wheel, agriculture, the arch, the plow, irrigation and many other innovations, and is often referred to as the Cradle of Civilization. The Sumerians developed the earliest known writing system - a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets - and this has meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early Egyptian mathematics. Indeed, we even have what appear to school exercises in arithmetic and geometric problems.

The Sumerian System, called "sexagesimal", combined a mundane 10...
with a "celestial" 6, to obtain the base figure 60. This system is in some
ways superior to our present one, and much superior to later Greek and
Roman systems. It enabled Sumerians to divide into fractions and multiply
into the million, to calculate roots or raise numbers several powers.

This was not only the first known mathematical system, but also one that
gave us... the "place" concept: Just as, (in the decimal system), 2 can be
2 or 20 or 200, depending on the digits place, so could a Sumerian 2
mean 2, or 120 (2 x 60), and so on, depending on the place.

The 360 degree circle, the foot and its 12 inches, and the "dozen" as
a unit, are but a few examples of the vestiges of Sumerian Mathematics,
still evident in our daily lives.

Their achievements in Astronomy, the establishment of a calendar, and
similar mathematical feats will come up later.

This idea of using position to arrange integers, known as the principle of position, is the first known use of such a system, the basis of our decimal system. This became lost until the fifth or sixth century CE, and western culture used the unwieldy Roman system of numbering, a tortuous and difficult system for performing math. Their system of numbering implies that they may have understood zero but, until further evidence is found, that remains largely conjectural.

The Sumerians, Babylonians and other inhabitants of the Euphrates valley certainly made some sophisticated mathematical advances, developing the basis of arithmetic, numerical notation and using fractions. Their work was adopted by the Greeks, and it is likely that the Greeks learned mathematical techniques from the Babylonian culture, as ideas traveled along the Silk Route from Anatolia (Turkey) to China. Alexander the Great is known to have sent astronomical records from Babylonia to Aristotle after he conquered the area.

Also, to represent the numbers 1 - 59 within each place value, two distinct symbols were used, a unit symbol () and a ten symbol () which were combined in a similar way to the familiar system of Roman< numerals (e.g. 23 would be shown as ). Thus, represents 60 plus 23, or 83. However, the number 60 was represented by the same symbol as the number 1 and, because they lacked an equivalent of the decimal point, the actual place value of a symbol often had to be inferred from the context.

Sumerians used their fingers.

Fingers, after all, are digits and underlie the digital economy. Our having ten of them, most of us, underlies the decimal system.

Each of your fingers has three distinct segments. and, touching now the middle segment of his right index finger with his right thumb, “Two.”

Now, still looking at your right palm, having successfully counted to 12, make a thumbs-up sign with your left hand.

As in . . . “that’s one set of 12.” Count another set of twelve with your right hand and you earn an unfolded left index finger (never mind that now your left hand is prepared to say, “bang-bang” – the Sumerians, gentle souls, had no guns). “That’s two sets of 12.”

Keep doing this until you have unfolded all five fingers of your left hand, and you’ve got 60.

One of the most ancient mathematical texts available is Plimpton 322. (1900 B.C)

Plimpton 322 is partly broken, approximately 13 cm wide, 9 cm tall, and 2 cm thick. New York publisher George Arthur Plimpton purchased the tablet from an archaeological dealer,Edgar J. Banks, in about 1922, and bequeathed it with the rest of his collection to Columbia University in the mid 1930s. According to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa.
The tablet is believed to have been written about 1800 BC, based in part on the style of handwriting used for its cuneiform script this handwriting is typical of documents from southern Iraq of 4000–3500 years ago." More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Plimpton 322 can be dated to the period 1822–1784 BC